addition and subtraction formulas for sine and cosine

Recall that a vector in ℝ2 can be rotated θ radians in the counterclockwise direction by multiplying on the left by the rotation matrix (cos⁡θ-sin⁡θsin⁡θcos⁡θ). Because rotating by α+β radians is the same as rotating by β radians followed by rotating by α radians, we obtain:

Note that sine is an even function and that cosine is an odd function, i.e. (http://planetmath.org/Ie) sin⁡(-x)=-sin⁡x and cos⁡(-x)=-cos⁡x. These facts enable us to obtain the subtraction formulas for sine and cosine.